= Solution
The function $x e^{-x}$ is strictly increasing on $(0,1)$ and strictly decreasing on $(1,\infty)$. Hence for each $\lambda>1$ there is a unique $\theta=\lambda^*\in(0,1)$ with $\theta e^{-\theta}=\lambda e^{-\lambda}$.
Apply the subcritical component estimates to $G(n,\theta/n)$. For a specified <vertex>, the exact <tree-component expectation in the Erdős-Rényi model> gives the limiting <probability> that it lies in a <tree component> of order $j$:
$$
a_j=\frac{j^{j-1}}{j!}\theta^{j-1}e^{-\theta j}.
$$
For each fixed $j$, the <probability> of a non-tree <graph component> of that size is $O_j(1/n)$: a <spanning tree> and one extra <edge> are necessary. Moreover the <Galton-Watson process> exploration estimate from the preceding proof bounds the <probability> of order greater than $K$ by $e^{-\delta_\theta(K-1)}$, uniformly in $n$, where $\delta_\theta=\theta-1-\log\theta>0$. First let $n\to\infty$ for fixed $K$, then let $K\to\infty$. No <probability> mass escapes to large components or non-tree components, so $\sum_{j\geq1}a_j=1$. Multiplying by $\theta$ and using its defining identity gives
$$
\boxed{\sum_{j=1}^\infty\frac{j^{j-1}}{j!}(\lambda e^{-\lambda})^j=\lambda^*.}
$$
Equivalently, the <rooted-tree generating function> selects the smaller solution of $T e^{-T}=\lambda e^{-\lambda}$. The choice of that branch is essential; the larger solution $\lambda$ is not the value of this convergent series.
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